UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering An Introduction to Pearl’s Do-Calculus of Intervention Marco Valtorta Department of Computer Science and Engineering University of South Carolina October 6, 2010

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 2 Contents Causal Bayesian networks The identifiability problem Graphical methods Completeness of Pearl’s do-calculus Summary and conclusion

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 3 Causal Bayesian Networks Causal Bayesian networks are Bayesian networks –Each variable in the graph is independent of all its non-descendents given its parents Causal Bayesian networks are causal –The directed edges in the graph represent causal influences between the corresponding variables

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Key reference Judea Pearl. Causality: Models, Reasoning, and Inference, 2 nd edition. Cambridge University Press, 2009 (First edition: 2000) 4

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Definition of Bayesian Network

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Visit to Asia Example Shortness of breath (dyspnoea) may be due to tuberculosis, lung cancer or bronchitis, or none of them, or more than one of them. A recent visit to Asia increases the chances of tuberculosis, while smoking is known to be a risk factor for both lung cancer and bronchitis. The results of a single chest X-ray do not discriminate between lung cancer and tuberculosis, as neither does the presence of dyspnoea [Lauritzen and Spiegelhalter, 1988].

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Visit to Asia Example α τ ε λ σ β δξ α (Asia): P(a)=.01ε (λ or β):P(e|l,t)=1 P(e|l,~t)=1 τ (TB): P(t|a)=.05 P(e|~l,t)=1 P(t|~a)=.01 P(e|~l,~t)=0 σ(Smoking): P(s)=.5 ξ: P(x|e)=.98 P(x|~e)=.05 λ(Lung cancer): P(l|s)=.1 P(l|~s)=.01δ (Dyspnea): P(d|e,b)=.9 P(d|e,~b)=.7 β(Bronchitis): P(b|s)=.6 P(d|~e.b)=.8 P(b|~s)=.3 P(d|~e,~b)=.1

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Intervention Causal models are useful because they represent many different models: –Each (hard) intervention creates a different model (“submodel” or “manipulated graph” [Spirtes et al., 2003]) Excision semantics capture the meaning of intervention Intervention variables allow for soft intervention 8

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Causality Matters! 9

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Experts Like Causal Models 10

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Visit to Asia Example Tuberculosis and lung cancer can cause shortness of breath (dyspnea) with equal likelihood. The same is true for a positive chest Xray (i.e., a positive chest Xray is also equally likely given either tuberculosis or lung cancer). Bronchitis is another cause of dyspnea. A recent visit to Asia increases the likelihood of tuberculosis, while smoking is a possible cause of both lung cancer and bronchitis [Neapolitan, 1990].

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 12 A Simple Smoking and Lung Cancer Model The simplest model X: Smoking, Y: Lung cancer X and Y are observable YX

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 13 Fisher’s Smoking and Lung Cancer Model Smoking-lung cancer model with genotype Genotype (U) is unobservable Fisher, R.A. “Cancer and Smoking.” Nature 182 (1958 August 30), pp X Y U

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 14 Smoking and Lung Cancer Model with an Intermediate Variable Smoking and lung cancer model with genotype and tar deposits in the lungs Tar deposits (Z) are observable X Y U Z

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 15 Cochran’s Soil Fumigants Example X: soil fumigants Y: oat crop yield Z 0 : last year’s eelworm population (unknown) Z 1 : eelworm populations before treatments Z 2 : eelworm populations after treatments Z 3 : eelworm population at the end of the season B: population of birds (unknown) We wish to assess the total effect of the fumigants on crop yield. But controlled randomized experiment are unfeasible, and Z 0 and B are unknown.

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 16 The Excision Semantics for Intervention The effect of an intervention that fixes the values of a (subset of the) variable(s) X to x is to remove the links getting into X and force X to have the fixed value(s) x The deletion of the link(s) into X represents the understanding that, whatever relationship existed between X and its parents prior to the action, that relationship is no longer in effect after we perform the action

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Effect of Interventions Theorem (adjustment for direct causes) Let PA i denote the set of direct causes of variable X i, and let Y be any set of variables disjoint of. The effect of the intervention on Y is given by where and represent pre-intervention probabilities.

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Imperfect Interventions 18

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Interventions as Variables The effect of an atomic intervention do(X i =x i ’) is encoded by adding to G a link F i  X i, where F i is a new variable taking values in {do(x i ’),idle}, x i ’ ranges over the domain of X i, and “idle” represents no intervention. Then we define:

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 20 Contents Causal Bayesian networks The identifiability problem Graphical methods Completeness of Pearl’s do-calculus Summary and conclusion

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 21 Identifiable Quantities The quantity P t (s) is identifiable if, given the causal Bayesian network G, the quantity P t (s) can be determined from the distribution of the observed variables P(n) alone –Alternative notation: For example, we may want to assess the total effect of the fumigants (t={X}) on yield (s={Y}), denoted P t (s), when we have the graph and P(X,Z 1,Z 2,Z 3,Y)

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 22 Unidentifiability Example(1) All the variables are binary. P(U=0) = 0.5, P(X=0|U) = (0.6,0.4), P(Y=0|X,U) = Y=0X =0X= 1 U = U= XY U

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 23 Unidentifiability Example(2) Note that We get: Because of the excision semantics, the link from U to X is removed, and we have: So, P X=0 (Y=0) = (0.7x0.5) + (0.2x0.5) = 0.45 X =0X= 1 Y =00.25 (=0.7x0.6x x0.4x0.5) 0.25 Y=10.25

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 24 Unidentifiability Example(3) All the variables are still binary. P(U=0) = 0.5 P(X=0|U) = (0.7,0.3) P(Y=0|X,U) = Y=0X =0X= 1 U = U= XY U

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 25 Unidentifiability Example(4) Using We still get: From We have P X=0 (Y=0) = (0.65x0.5) + (0.35x 0.5) = 0.4 <> 0.45 So, P X (Y) is unidentifiable in this model X =0X= 1 Y =00.25 Y=10.25

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 26 Identifiability Problem For a given causal Bayesian network, decide whether P t (s) is identifiable or not If P t (s) is identifiable, give a closed- form expression for the value of P t (s) in term of distributions derived from the joint distribution of all observed quantities, P(n)

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 27 Contents Causal Bayesian networks The identifiability problem Graphical methods Completeness of Pearl’s do-calculus Summary and conclusion

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 28 Back-door criterion Front-door criterion The three “do-calculus” inference rules Causal effect with single intervention variable Extensions of back-door criterion Extensions of front-door criterion Graphical Methods

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 29 Back-Door Criterion A set of variables Z satisfies the back-door criterion relative to an ordered pair of variables (X i, X j ) in a DAG G if (1) no node in Z is a descendant of X i, and (2) Z blocks every path between X i and X j that contains a directed link into X i Similarly, if X and Y are two disjoint subsets of nodes in G, then Z is said to satisfy the back-door criterion relative to (X,Y) if it satisfies the criterion relative to any pair (X i, X j ) such that X i  X, X j  Y

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 30 Use of the Back-Door Criterion If a set of variables Z satisfies the back- door criterion relative to (X, Y), then the causal effect of X on Y is identifiable and is given by the formula:

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 31 Using the Back-Door Criterion: an Example We can use the back-door criterion on the simple lung cancer model Here Z =ø, and we get: YX

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 32 Smoking and the genotype theory X: Smoking Y: Lung Cancer Z: Tar Deposits Can we get ?

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 33 Effect of Smoking on Lung Cancer in the Presence of Tar Deposits We compute in two steps: the causal effect of x on z and the causal effect of z on y. By application of the backdoor criterion with an empty set of concomitants, And, by the backdoor criterion with X as a concomitant, Putting things together:

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 34 Front-Door Criterion A set of variables Z is said to satisfy the front-door criterion relative to an ordered pair of variables (X, Y) if (1) Z intercepts all directed paths from X to Y (2) there is no back-door path from X to Z (3) all back-door paths from Z to Y are closed by X

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 35 Use of the Front-Door Criterion If Z satisfies the front-door criterion relative to (X,Y) and if P(x,z) >0, then the causal effect of X on Y is identifiable and is given by formula: The front-door criterion may be obtained by a double application of the back-door criterion, as shown in the tar deposits example

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 36 Using the Front-Door Criterion: an Example In the smoking-lung cancer model with genotype and tar, we may use the front-door criterion and get: X Y U Z

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 37 Notation for Graphs with Link Deletion In a causal Bayesian network G, Let X, Y, Z be arbitrary disjoint set of nodes : the graph obtained by deleting from G all directed links pointing to nodes in X : the graph obtained by deleting from G all directed links emerging from nodes in X : the deleting of both incoming and outgoing links

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 38 Pearl’s Calculus of Interventions For any disjoint subsets of variables X, Y, Z, and W: Rule 1 (Insertion/Deletion of Observations): Rule 2 (Action/Observation Exchange): Rule 3 (Insertion/Deletion of Actions): –Where Z(W) is the set of Z-nodes that are not ancestors of any W-nodes in

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 39 Keshan Disease To determine the causal effect of Selenium on Keshan Disease, we need to find a set Z of variables (set of concomitants) that satisfies the back-door criterion Z={ Region of China} is an answer. Z={Genotype} would also be an answer if it were observable. Backdoor Path

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 40 The Smoking Example.1 Based on rule 2, we have

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 41 The Smoking Example 2 Note: we can use the same process to prove the back-door formula

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 42 The Smoking Example 3 Based on 1,2, and above, we get: Note: this is also a proof of the front-door formula

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 43 The Smoking Example 4 See 1 and the third formula in 3; we have:. 5 See 2

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 44 Some Nonidentifiable Models

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 45 Some Identifiable Models

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 46 Why They are Identifiable (a), (b) rule 2, (c), (d) back-door (e) front-door (f) (g)

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 47 Why g? Z1 block all directed paths from X to Y Z2 blocks all back- door paths between Y and Z1 in Putting the pieces together, we obtain the claimed result

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 48 Several More Nonidentifiable Models

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 49 Derived Algorithms and Criteria The back-door and front-door criteria can be obtained by using the three inference rules The algorithm and derived criteria listed below can also be obtained by using the three rules –Causal effect with single intervention variable (Galles & Pearl) –Extension of Back-door criterion (Robins & Pearl) –Extension of Front-door criterion (Kuroki & Miyakava) For the causal effect of X and Y, where X and Y are individual variable, Tian proved a beautifully simple necessary and sufficient criterion [Thm Pearl 2009]

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 50 Contents Causal Bayesian networks The identifiability problem Graphical methods Completeness of Pearl’s do- calculus Summary and Conclusion

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 51 Pearl’s Conjecture Pearl conjectured that P t (s) is identifiable if and only if all terms involving intervention can be replaced by terms involving only observations, by successive application of the three inference rules of the “do calculus”

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Completeness of the do-calculus Various papers by Tian, Shpitser, Pearl, Huang, Valtorta Yimin Huang and Marco Valtorta. "Pearl’s Calculus of Intervention is Complete." Proceedings of the 22nd Conference on Uncertainty in Artificial Intelligence (UAI- 06), Cambridge, MA, July 13-16, 2006, pp Yimin Huang and Marco Valtorta. "Identifiability on Causal Bayesian Networks: A Sound and Complete Algorithm." Proceedings of the Twenty-First National Conference on Artificial Intelligence (AAAI-06), Boston, MA, July 16-20, 2006, pp The paper above extends to general causal Bayesian networks the results in: Yimin Huang and Marco Valtorta. "On the Completeness of an Identifiability Algorithm for Semi-Markovian Models." Annals of Mathematics and Artificial Intelligence, 54, 4, , 2009 The dates are backwards because of publication delays 52

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 53 Pearl’s Graphical Inference Rules Are Complete Exploiting the complete Identify algorithm on normal causal Bayesian networks, we prove that Pearl’s three inference rules are complete This result confirms Pearl’s conjecture

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 54 Outline of the Proof The sound and complete algorithm for computing P T (S) is obtained by using lemma 1 and lemma 2 We show that these two lemmas can be obtained by just using the three inference rules and standard probability manipulations. Therefore, the sufficiency of the three rules is proved Three inference rules Tian & Pearl’s two lemmas A sound & complete algorithm

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 55 Contents Causal Bayesian networks The identifiability problem Graphical methods Completeness of Pearl’s do-calculus Summary and Conclusion

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering 56 Summary: Algorithms for the Identifiability Problem in Causal Bayesian Networks Each variable is independent of all its non-descendents given its direct parents The directed edges in the graph represent causal influences between the corresponding variables Some variables (e.g., genotype) are not observable Can we identify a causal effect from observable data alone? –E.g., can we compute the probability of lung cancer given smoking in the three models of the left, even when the genotype is not observable? Are there efficient, sound, and complete algorithms to solve the above identifiability problem? SmokingLung Cancer Genotype Lung CancerSmoking Genotype Smoking Lung Cancer Tar Deposits

UNIVERSITY OF SOUTH CAROLINA Department of Computer Science and Engineering Slides available at 57